Current correlations, Drude weights and large deviations in a boxball system
Abstract
We explore several aspects of the current fluctuations and correlations in the boxball system, an integrable cellular automaton in one space dimension. The state we consider is an ensemble of microscopic configurations where the box occupancies are independent random variables (i.i.d. state), with a given mean ball density. We compute several quantities exactly in such homogeneous stationary state: the mean value and the variance of the number of balls N _{ t } crossing the origin during time t, and the scaled cumulants generating function associated to N _{ t }. We also compute two spatially integrated currentcurrent correlations. The first one, involving the longtime limit of the currentcurrent correlations, is the socalled Drude weight and is obtained with thermodynamic Bethe ansatz (TBA). The second one, involving equal time currentcurrent correlations is calculated using a transfer matrix approach. A family of generalized currents, associated to the conserved charges and to the different time evolutions of the models are constructed. The longtime limits of their correlations generalize the Drude weight and the second cumulant of N _{ t } and are found to obey nontrivial symmetry relations. They are computed using TBA and the results are found to be in good agreement with microscopic simulations of the model. TBA is also used to compute explicitly the whole family of flux Jacobian matrices. Finally, some of these results are extended to a (noni.i.d.) twotemperatures generalized Gibbs state (with one parameter coupled to the total number of balls, and another one coupled to the total number of solitons).
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 June 2022
 DOI:
 10.1088/17518121/ac6d8c
 arXiv:
 arXiv:2201.13126
 Bibcode:
 2022JPhA...55x4006K
 Keywords:

 boxball system;
 Drude weight;
 integrability;
 generalized hydrodynamics;
 thermodynamic Bethe ansatz;
 large deviation;
 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 v2: minor changes. 41 pages, 9 figures. To a appear in J. Phys. A